This application relates to measurement and analysis of stresses in devices with features fabricated on substrates.
Substrates formed of suitable solid-state materials may be used as platforms to support various structures, such as large panels with layers or coatings formed thereon and microstructures integrated to the substrates. Examples of large panels include, among others, aeronautical and marine components and structures. Examples of substrate-based integrated devices with one or more layers or films include, among others, integrated electronic circuits, integrated optical devices, micro-electro-mechanical systems, flat panel display systems, or a combination of two or more of the above devices.
In the above and other structures, components, and devices, different materials or different structures are usually formed on the same substrate and are in contact with one another. Some devices may also use complex multilayer or continuously graded geometries. Hence, the interfacing of different materials and different structures may cause a complex stress state in each feature due to differences in the material properties and the structure at interconnections under different fabrication condition and environmental factors (e.g., variations or fluctuations in temperature). In fabrication of an integrated circuit, for example, the stress state of the interconnect conducting lines may be affected by film deposition, rapid thermal cycling, chemical-mechanical polishing, and passivation during the fabrication process. Stresses caused by these and other factors may adversely the performance and reliability of the devices and may even cause device failure.
Hence, measurements and analysis of changes in stresses and deformation of a substrate and features fabricated on the substrate may have important applications in various industrial areas. For example, it is desirable to measure stresses on various features formed on the substrate to improve the design of the device structure, selection of materials, fabrication process, and other aspects of the devices so that the yield, device performance, and device reliability of the device can be enhanced. The stress measurements may be used to assess or evaluate the reliability of materials against failure from such phenomena as stress migration and electromigration, stress-voiding and hillock formation. The stress measurements may also be used to facilitate quality control of the mechanical integrity and electromechanical functioning of circuit chip dies during large scale production in wafer fabrication facilities. In addition, the stress measurements may be used to improve the design of various fabrication processes and techniques, such as thermal treatments (e.g., temperature excursions during passivation and annealing) and chemical and mechanical treatments (e.g., polishing) to reduce residual stresses in the final device.
This application includes techniques for determining large deformation of layered or graded structures to include effects of body forces, such as gravity, electrostatic or electromagnetic forces, loading or supporting forces, and other forces that uniformly distributed over the structures, and effects of forces that concentrate at certain locations of the structures.
In one embodiment, the technique includes the following steps. A plate structure formed of one or more materials is used to represent a device which may have one or more discrete layers, a continuously graded structure, or a combination of both. Each material is assumed to exhibit linear elastic deformation. A first spatial-varying function, that is uniform within a plane of the plate structure and varies along a direction perpendicular to the plane, is used to represent a body force acting on the device which affects evolution of curvature of the device. A second spatial-varying function, that is uniform within the plane and varies along a direction perpendicular to the plane and with a temperature of the device, is also used to represent effects of thermal stresses in the device. Nonlinear functions of positions within the plane are further used to represent displacements of the device within two principal directions within the plane and a third principal direction perpendicular to the plane, respectively, to include effects of large deformation.
Next, a total potential energy of the device is computed based on the first and the second spatial-varying functions and the nonlinear functions for the displacements. The total potential energy are then minimized with respect to principal curvatures and axial stretches respectively along the two principal directions within the plane to derive analytical relationships between an effective force for causing curvature evolution of the device and principal curvatures along the two principal directions.
The above analytical method may be combined with a technique for measuring curvatures of surfaces to determine stresses in devices with a plate structure including multi-layered devices. For example, an optical coherent gradient sensing (CGS) method may be used to provide full-field, real-time, and noninvasive measurements of curvatures of reflective surfaces to analysis under the large deformation analytical method. A stress monitoring system may be constructed based on the combination to provide in-situ monitoring during fabrication of devices such as flat panels and semiconductor circuits.